Environmental Engineering Reference
In-Depth Information
where Φ is the cumulative distribution function of the standard normal distribution. Suppose
that
n
proof tests are conducted and none of the test piles fails at
x
T
. If the standard devia-
tion, ξ, of ln(
x
) is known but its mean, μ or η, is a variable, then the probability that all of
the
n
test piles do not fail at
x
T
is
n
ln
()
x
−
η
∏
1
n
T
L
()
µ
=
p
(
xx
≥
)
=
Φ
−
(14.3)
X
T
µ
ξ
i
=
L(μ) is also called “likelihood function” of μ. Given L(μ), the updated distribution of the
mean of the bearing capacity ratio is (e.g., Ang and Tang 2007)
fln(
x
−
η
n
T
f
′′
()
µα
=
Φ
−
f
′
()
µ
(14.4)
ξ
where
f
′(μ) is the prior distribution of μ, which can be constructed based on the empirical
log-normal distribution of
x
and the within-site variability information (Zhang 2004), and
α is a normalizing constant:
−
1
∞
fln(
x
−
η
∫
Φ
n
(14.5)
α
=
−
f
′
()d
µµ
ξ
−∞
Given an empirical distribution
N
(μ
X
, σ
X
), the prior distribution can also be assumed as a
normal distribution with the following parameters:
′ µ
X
(14.6)
2
2
(14.7)
σ
′ =
σ
−
σ
X
The updated distribution of the bearing capacity ratio,
x
, is thus (e.g., Ang and Tang 2007)
∞
∫
fx
()
=
fx f
(
µµµ
)
′′
()
d
(14.8)
X
X
−∞
where
f
X
(
x
|μ) is the distribution of
x
given the distribution of its mean. This distribution is
assumed to be log-normal as mentioned earlier.
14.3.2 Proof load tests that do not pass
More generally, suppose only m out of n test piles do not fail at
x
=
x
T
. The probability that
this event occurs is
n
m
=
m
(
nm
−
)
−≥
L
()
µ
px
(
≥
x
)
1
p xx
(
)
(14.9)
T
µ
T
µ
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